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单词 InequalityWithAbsoluteValues
释义

inequality with absolute values


Recalling that the absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/AbsoluteValue) of a real number means on the number line (real axisMathworldPlanetmath) the distance of the point from the origin, we have the following three rules which remove the absolute value signs from an inequality (we use the logical symbol “” for alternativeness ‘or’).  Note that the symbols “” and “” may also be without the equality bar.

  1. 1.

    |a|b  a-bab

  2. 2.

    |a|b  -bab

  3. 3.

    |a||b|a2b2

These rules are valid for all real values of a and b.  For example, if one has a case

|x|<-5

corresponding the rule 2, this inequality seems to be impossible since no absolute value is negative; but now also the result  -(-5)<x<-5  given by the rule 2 is impossible — no real number is simultaneously greater than +5 and less than -5.

Examples.  We solve some inequalities with absolute values.

a)  |2x+1|>5x
2x+1<-5x  or  2x+1>5x   (rule 1)
7x<-1  or  -3x>-1
x<-1/7  or  x<1/3
x<1/3   (combined)

b)  8|x|+|x-2|>6
|8x|>6-|x-2|
8x<-6+|x-2|  or  8x>6-|x-2|   (rule 1)
|x-2|>8x+6  or  |x-2|>6-8x
x-2<-8x-6  or  x-2>8x+6  or  x-2<-6+8x  or  x-2>6-8x  (rule 1 twice)
9x<-4  or  -7x>8  or  -7x<-4  or  9x>8
x<-4/9  or  x<-8/7  or  x>4/7  or  x>8/9
x<-4/9  or  x>4/7   (from the number line)

c)  |1-5x|3
-31-5x3   (rule 2)
-4-5x2   (subtracted 1 from all parts)
4/5x-2/5   (divided by -5)
-2/5x4/5   (rewritten from end to begin)

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更新时间:2025/5/4 16:09:28